Three-pion contribution to hadronic vacuum polarization

Abstract

We address the contribution of the 3π channel to hadronic vacuum polarization (HVP) using a dispersive representation of the e + e − → 3π amplitude. This channel gives the second-largest individual contribution to the total HVP integral in the anomalous magnetic moment of the muon (g − 2) µ , both to its absolute value and uncertainty. It is largely dominated by the narrow resonances ω and φ, but not to the extent that the off-peak regions were negligible, so that at the level of accuracy relevant for (g − 2) µ an analysis of the available data as model independent as possible becomes critical. Here, we provide such an analysis based on a global fit function using analyticity and unitarity of the underlying γ * → 3π amplitude and its normalization from a chiral low-energy theorem, which, in particular, allows us to check the internal consistency of the various e + e − → 3π data sets. Overall, we obtain a 3π µ | ≤1.8 GeV = 46.2(6)(6) × 10 −10 as our best estimate for the total 3π contribution consistent with all (low-energy) constraints from QCD. In combination with a recent dispersive analysis imposing the same constraints on the 2π channel below 1 GeV, this covers nearly 80% of the total HVP contribution, leading to a HVP µ = 692.3(3.3) × 10 −10 when the remainder is taken from the literature, and thus reaffirming the (g−2) µ anomaly at the level of at least 3.4σ. As side products, we find for the vacuum-polarization-subtracted masses M ω = 782.63(3)(1) MeV and M φ = 1019.20(2)(1) MeV, confirming the tension to the ω mass as extracted from the 2π channel.

g−2 of charged leptons, α(MZ2) , and the hyperfine splitting of muonium

“…A recent study of the three-pion contribution to the hadronic vacuum polarization based on a global fit function using analyticity and unitarity constraints [38] highlighted major differences arising in various determinations of a π þ π − π 0 μ . These were attributed to the choice of cross section interpolation used in the prominent ω resonance region when integrating the data.…”

“…[1] larger than that found in Refs. [37,38]. In order to address this issue in this work, the clusters and covariance matrix elements corresponding to the fitted ω resonance alone have been interpolated to a 0.2 MeV binning using a quintic polynomial.…”

“…In the π þ π − π 0 channel, adjustments have been made to the treatment of the narrow ω resonance, which is now integrated over using a quintic polynomial interpolation in order to avoid an overestimation of the cross section from a linear interpolation that was recently noted in Ref. [38]. This has reduced the mean value of a π þ π − π 0 μ by ∼1 × 10 −10 and, in turn, contributed to a significant reduction of the mean value of a SM μ in this work, although it is important to note that all estimates from this analysis are consistent with those given in Ref.…”

Hadronic Vacuum Polarization:(g−2)μversus Global Electroweak Fits

“…Traditionally, hadronic vacuum polarization (HVP) has been determined via a dispersion relation from the cross section σðe þ e − → hadronsÞ [ where in the usual conventions for isospin-breaking effects the integral starts at the threshold s thr ¼ M 2 π 0 due to the e þ e − → π 0 γ channel [52] and the kernel functionKðsÞ can be expressed analytically. Global analyses based on a direct integration of cross section data [25,26,29,30] can now also be combined with analyticity and unitarity constraints for the leading 2π [27,29,53] and 3π [28] channels, covering almost 80% of the HVP contribution, to demonstrate that the experimental data sets are consistent with general properties of QCD, and radiative corrections for the 2π channel have been completed at next-to-leading order [54]. With recent advances in constraining the contribution from hadronic light-by-light scattering (including evaluations [33][34][35]37,38,[55][56][57] based on dispersion relations in analogy to Eq.…”

“…(1), short-distance constraints [39][40][41], and lattice QCD [36,42]) as well as higher-order hadronic corrections [30,31,43,58], this data-driven determination of HVP has corroborated the ðg − 2Þ μ tension at the level of 3.7σ. Nevertheless, since by far the largest hadronic correction arises from HVP, requirements for the relative precision are extraordinary, with a HVP μ ¼ 693.1ð4.0Þ × 10 −10 [20, [25][26][27][28][29][30] as currently determined from e þ e − → hadrons cross sections corresponding to less than 0.6%. One may thus ask what would happen if the SM prediction were brought into agreement with experiment by changing a HVP μ .…”